Finding Transition Pathways on Manifolds

When a randomly perturbed dynamical system is subject to some constraints, the trajectories of the system and the noise-induced most probable transition pathways are restricted on the manifold associated with the given constraints. We present a constrained minimum action method to compute the optimal transition pathways on manifolds. By formulating the constrained stochastic dynamics in a Stratonovich stochastic differential equation of the projection form, we consider the system as embedded in the Euclidean space and present the Freidlin--Wentzell action functional via large deviation theory. We then reformulate it as a minimization problem in the space of curves through Maupertuis principle. Furthermore we show that the action functionals are intrinsically defined on the manifold. The constrained minimum action method is proposed to compute the minimum action path with the assistance of the constrained optimization scheme. The examples of conformational transition paths for both single and double rod molecules in polymeric fluid are numerically investigated.